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On the transition phase in robotics- Part 2: Control
Preprints of the Fourth IFAC Symposium on Robot Control September 1~21 , 1994, Capri , Italy
On the transition phase in robotics- Part 2: control B.Brogliato, P. Orhant, R. Lozano* * For the adress of authors please see the companion paper [2]
1
Introduction
This note is devoted to study the stabilization of robot manipulators during complete robotic tasks, i.e. tasks involving both constrained and non-constrained motion phases. It follows [1] [2] where we have stud ied impacts models and dynamics since such tasks invariably imply impacts phenomenons. This kind of problems have received few attention in the robots control literature; among them an interesting work can be found in [5] where the environment is assumed to be compliant. One of our first motivations in doing this work comes from [6], where R.P. Paul stated that The contact problem is unsolved for rigid manipulator, rigid sensor, rigid environment problems. We hope that this work is a first answer. The note is organized as follows : In section 2 we formulate the problem and we study a simple strategy. In section 3 we study more complex controllers that involve switches, and we propose a framework for stabilization of such systems. Conclusions follow.
2
On the control of a "complete" robotic task
• Problem formulation. The problem of the complete robotic task control can be represented by the following set of equations: non-contact : impact:
contact:
. _ (8.) : Z = gne(z, ud, z(O) -
Zo
(t;) the impact; we assume that the environment imposes kinematic constraints on the manipulator that can be written as (z) = O. (8.J represents the motion of the robot's tip along the environment's surface and the dynamic (or algebraic) force error equation, and (8... ,) represents the constrained equation in some suitable coordinates; y(.) may be composed of some components of the task-space velocity X, or of the generalized velocity q. The "high-level control strategy" H LC8 schedules the switches between control laws U ne E U n e and U c E U c (i.e. U = Ui if certain conditions Ci are true. Roughly Zm is the normal component of Z to the constraints. In (1), z(t) may be composed of generalized position and velocity q, q, or of Cartesian coordinates X, X. Then (8.J and (8.,J may be expressed in transformed generalized coordinates, or in task space coordinates [9].Note that the dynamic behaviour of the system considering (8.) and (8 y ) may be quite different from the behaviour considering separately (8.) or (8y ). These difficulties are illustrated in this section. The analysis proceeds in two steps: firstly we study the stability of the bouncing phase, then we examine stability of the constrained phase (if the bouncing phase has occured stably): By stability we mean that the system's state does not diverge and that the robot's tip is eventually at rest on the surface . • computed torque control. Consider the dynamic equations of a rigid manipulator. Let us denote X the vector of position and orientation of the end-effector. Then X = X(q) and X = J(q)q. The generalized forces working on X acting on the end-
. ( (;) effector are impulsive, i.e. A = pOt.. Assuming that (8.) : z =+gne z, Ui, _t). J(q) is square and full-rank, we can apply the control law y y(t k ) = £(y(t k ,phys. para~'b ]Tv, with v = -MJjq + J-TC(q,q)q + J-Tg(q) + { (8 ): . (8.~): Zm = 0 M J (-A2 X -AI(X -Xd )), where M J = J-TMJ-l, Xd is With Y = y(z) the desired value of X . The closed-loop equation is given by: (2)
. . u, = u,(z, x, t, ze) With x = h(z , x , t, control: 1 U, E UncaT"U, E UC { HLC8:u = u, ifC, is true (1) The differential equations (8.) are the dynamical equations for the free motion of the system with and without impacts; the difference equation (8 y ) represents algebraic equations relating the variable value after and before
(tn
Thus we get: zc)
{X;} + A2 X i + Al (Xi - X id )
=0
(3)
where X, denotes the t'th component of X , and: (4)
We now assume that the environment is represented by the kinematic constraints (X) = 0, E lR m , m :S 6.
571
Following Yoshikawa's ideas [9], we split X into two parts as EX = [ EpX. ]
E,X
[
x.P ]. X,
Then the constrained
equation is X, = X, = O. E = [
~;
] is thus a full-
rank matrix that transforms X into task-frame velocity EX. Assume that E = 0 (i.e. the constraint has constant normal vector), i.e. (X) = E,X with E, constant. Then we can write:
A similar analysis can be done when time-varying desired trajectories are considered. However stability may fail in this case since the transition phase including bounces is less easy to stabilize. This indicates the limitations of using a single control law in general. Notice that from EX = EJq and EX(tt) =
Remark. co EX' (t-) k
. Wit
C
h,,= (In0- m -elm 0) co
LI.
duce that (Sy) is Q(tt) (EJ)-I[EJ so that [j with Mj = E- T MjE- 1 = MJ > O. Clearly only X, (the normal component of EX to the surface) has discontinuities due to the impacts, since the surface is supposed to be fri ctionless. Following [4] and since we assume the Jacobian to be full-rank (i.e. the last link is allowed to move in any direction) we deduce that (Sy) is represented . + . . + . - I by X,(tk) = -eX,(tk)' Xp(t k ) = Xp(t k ) . Now the problem is to see whether IX,(t;+I)1 :S IX,(tt)!. Between two impacts we get from (5):
and:
x, + )..2X, +
)..1 (X,
- X'd) = 0
(7)
Thus it can be shown that X, strictly decreases between two impacts (X,(tk+l) = X,(tk) = 0 for all k) and we inwhereas fer that X, and X, tend to zero in finite time XI' and XI' remain bounded during the bouncing phase. (we assume that Xd is such that E,Xd = X,d is "inside " the environment). Thus the impact phase is stable. Let us now examine the closed-loop constrained system: just after and as long as contact is maintained we get:
t"
t,
(8) where Fe is the interaction force expressed in the taskframe coordinates, Fe
= ( F~,
) during constrained
motion, El = rh-m 0], E2 = [0 Im]. Thus Fe! = _(~M;IE'[)-I)..IX!d, and we deduce that all closedloop signals are bounded. Note that the control law in [9] decouples closed-loop dynamics, whereas the computedtorque controller introduces a "disturbance" term in the free-motion equation for XI" It can also be proved that a P D controller does not only guarantee global stability in free motion regulati(m, but also ensures boundedness of the state and the interaction force when the robot collides a rigid environment. 1 Notice
that we are not able to deduce which components of are discontinuous from (4) alone. We need further arguments like the fact that the surface is friction less and Newton's restitution coefficient law . Thus (4) is to be used to compute p from q X rather than the inverse.
EX
(Inom
(-1
)~en Im
ergy is conserved and e <
[jq(t;), with [j (EJ)-I[nEJ, with [n
deLI.
). Clearly whel ' :: = 1 kinetic en-
Iq(tnl
= Iq(t;)I.
1 [n (Inom 0:) then
we can
--+
Notice that if
which indicates that
the bouncing phase (Sy) considered as a discrete time system is convergent. 3
Switching control strategies
3
We now deal with thecase when several control laws are alternatively switched depending on whether the robot is constrained or not. Before going on with a simple example, let us note that when the controller switches from Ui to Uj, the control input may not be defined at the switching time, which in turn may be a switching period. During this time c;>r interval, a reasonnable assumption is that U E [um, UM], where U m and UM may also be time and state dependent. Then the equations in (S%) and (SzJ are differential inclusions. Basic facts and terminology about differential inclusions may be found in [3].
3.1
A simple example
Consider a mass m moving on a horizontal plane with a control force U and a rigid obstacle. Assume that the materials are such that 0 < e < 1. We assume that the controller can a priori switch between three values: • U nc = ->"2 X - >"1 (x - Xd)
A5(Fe - Fd) + >"4 f;o (Fe - Fd)dr - Fe Fd) - Fe with Xd > 0, Fd < 0, and To denotes the switching time from another controller to u c . • Uc
=
•
= >"3(Fe -
Ut
The dynamical equations of the system are given by mx = U in free space, mx = U + it(t) during the impact phase, Fe + U = 0 during the constrained phase. Let us proceed as in [2] to transform the system with distributions into a Caratheodory system. Let us define the "impact" function h(t) where h is of bounded variation on the interval of impacts and it represents the impact impulsive forces (the derivative being taken in distributional sense) and f(t) such that f is a .e. differentiable but at the points of discontinuities of h, with j = h outside these points. Since changes of variables
572
To < tf (Le. the bounces have not stopped), and that To coincides with an impact. The closed-loop equation is:
are allowed in differential inclusions [3] (p.99, th.l), we set Y
=(
~~
) =(
~
{m ).
) - (
Then we get:
mi (
~~
) = (
~
)
+(
~
{~~~
A4 J~ Fddt
= -A5Fd -
A4Fdt + h(t)
+ h(t) (10)
)
(9)
Thus we have transformed the original system with distributions in coefficients into a differential inclusion of the form iJ E F(y, u) = Ay + Bu, with U E [um(y, t), UM(y, t)] at the switching times, U given by one of the controllers above between two switches. We can consider um(y, To) = min(u(To-), u(To+)), UM(y, To) = max(u(To-), u(To+)). The local existence and uniqueness theorems in [7] for the nonautonomous case require that the multifunction be measurable in t . We deduce that provided both functions um(Y, t) and UM(y, t) are measurable in t, the general theorems above can be applied to (9) for any initial data outside the discontinuities of h(t) (i.e. they can be applied almost everywhere with h as above). Note that if the solutions y(t) to the inclusion in (9) exist and are unique (this is to be understood for reachable sets) on a certain interval [to, t], so do the original trajectories X(t) = y(t) - (
= -A5Fd -
).
• TI-ansition phase stability with H LC5} Let us consider our example. Assume we choose the following strategy H LC5}: After the first impact has occured switch to Ut (If the mass is still bouncing, then Ut = -A3Fd). Then after a certain prespecified time switch to U c . Notice that we may assume in a first study that the switches occur infinitely fast. Now we can reasonnably assume that the bounds um(Y, t) and UM(Y, t) are measurable in t, because each time a switch occurs, both controllers before and after the switch lead to an equation in (9) with a time-measurable RHS, since Unc = unc(y}.!(t)) when the switch between Unc and Ut occurs. At a given time t, either one of the above controllers is used, in which case F(t, y) is a point in the plane and (9) is an ordinary differential equation for which Carath60dory conditions trivially holds, or t corresponds to a switching instant, and F(t, y) is a segment, thus convex. The only case of interest is when a transition between U nc and another controller occurs, since otherwise Urn and UM can be assumed to be independent of y. In this case one can verify that o:(F(t,y),F(t,y')) has the form «Y2 - y;)2 + (UM - u~ )2)i, where UM can also be replaced by U m depending on the values of the bounds. Thus o:(F(t,y),F(t,y')) ~II y - y' 11 provided the bounds are smdoth enough functions of y, so that theorem 2.2 in [7] applies. Then if we prove that no finite escape can occur in the system, the solutions (Le. the reachable sets) from any admisssible initial data exist and are unique on [0, +00). The controllers Unc and Ut both lead a stable transition phase. Assume now that for some reason U c is switched at t = To (we take To = 0 to simplify the analysis), with
Between two jumps we get mi = "-A5Fd - A4Fdt, or i = ·at + b with a = _A.Fd > 0, b = -~ > o. Thus m m ± = + bt + c, with c = ±(O+) < o. Note that ± is increasing and ±(t) > 0 for some t, so that the mass collides the wall again at the time t} = ~ ( -~ + ~),
af
t!. = ~ - ~a±(O+) > 0, and ±(t;-) = ~ti + bt} + ±(O+) > ±(O+). Note that t} -+ as a -+ o. We want that 1±(ti)1 = e±(t;-) < I±(O+)I· This condition can be written as ~ti+bt} < I±(O+)I~. When A4 -+ 0, it becomes simply e < 1. We deduce that there exists A4 > 0 such that for 0 < A4 < A4 the condition is fulfilled. We have asssumed that To coincides with a percussion, so that x(O) = 0 and we can explicitely calculate t}. Note however that if such is not the case, we can still assert existence of tJ (because there exists tJ such that for t > tJ, ±(t) > 0), and of A; as it can be shown that A4 = 0 implies a stable transition phase (for any A5 > 0, there exists A4 > 0 such that A4 E (0, A4) yields 1±(ti)1 < I±(O+)I) . Now we can repeat this reasoning for intervals [tJ, t2], etc... by setting each time tk = 0 and integrating between tk and tk+l. It can also be proved that the values of A4 on the successive intervals are in some sense independent of k. We conclude that there exists A4 > 0 such that the impact phase with U = U c is stable. There is a clear reason why adding an integral force feedback introduces some difficulties in the analysis of the transition phase behaviour. In effect, by combining (5z ) and (511 ) when Ut is used we are able to get the following recurrence equations for ±(tk): ±(tt+J) =
-¥
-e [29a
(-~ + ~r + ~ (-~ +~) + ±(tn] ,
for
which 0 si an equilibrium point. The only thing that we have been able to conclude is that for A4 small enough, these recurrence equation lead for any initial condition to t!.k -+ 0 and :i:(tk) -+ 0 as k -+ +00. • TI-ansition phase stability with H LC52 We assume now that the switch between U nc and Ut is triggered when x(t) = -j..I. ~ 0, and U c is switched on when F. > o. We assume that the force sensor response to impacts is such that after each percussion, U c is switched on during a certain period of time t!.c. Then it can be shown that there exists A4 > 0 such that for o < A4 < A4, H LC52 leads to a stable task whatever t!.c may be.
3.2
A Lyapunov-like stability analysis
In view of the extension of the preceding results to the case of n-dof manipulators, it is interesting to estab-
573
lish stability using Lyapunov functions V. First notice that since we have guaranteed existence of trajectories (see [2]), Lyapunov arguments can be used to prove stability. Roughly, when a jump occurs in the solution, there corresponds a jump .6.v(tk) in V . We thus have to analyze the continuous variation and the discontinuous variation of V. We shall require the use of a single Lyapunov V function to prove stability of a whole task as described in (1) . The main difficulty will be to guarantee negative V and ~ V (tk) along all trajectories of (1) , knowing that at certain times, a "wrong" controller may be applied. Let us for the moment investigate the one-dof case, and consider the function
(T2k+2, tk"+I) and (tt, T 2k+d respectively), and since ~ViT"+2,T,.+d 5 0 we deduce that V(tt+l) 5 V(tt). In conclusions, V is negative definite during the free motion 5 V(To) and V(tt+l) 5 V(tt) during the phase, V(t impact phase so that V (t f) :S V (To) (t f is the time when bounces stop, see example 6 above), and Fe = Fd < 0 for
o)
t 2: tf· Another strategy H LC 53 may be to apply Ut for all t 2: To. In this case, we only have to verify that between impacts V decreases. We get: V(tk"+I) - V(tt) = -),3Fd [x(t)]::+1 +),1 [x(t)21:: +1 +(C-),2) I/:+I ±2(t)dt-
e>-;';d I/:+l x(t)dt 5 0 since XFd > 0 on [tt, tk"+I] and >'2 > C. V = ~m±2 + (x - Xd)2 + c:i:x + ~ (J~ Fe(T)dT • integml force feedback with c > 0 satisfying c2 < m),I ' Suppose that ),4 = 0, We now briefly analyze the case when ),4 > 0, i.e. there is ),5 = ),3, Ut = ),3(Fe - Fd) - Fe - ),2± (we add dampsome integral action in the force feedback, and the strating in the transition control law 2) and that H LC5 = egy consists of switching between Une and Ue at To. We do H LC52 . Along trajectories of (5z ) with U = U ne we get the hypothese that Fd == 0 whenever U i= U e . Thus along V 5 (->'2 +c+ ~)±2 + (-~c +~) x2 , thus V is neg- trajectories of (5z ) with U = Une, V is negative definite ative definite for ),2 > C + ~ , ),1 > ~ . We implicitely in X, ± since I~ FddT is constant during that time. The assume that the impact phase effectively occurs, i.e. {td first variation of V to analyze is that at the impact times; 2 is an infinite sequence. Then a sufficient condition for the we get: ~ V(tk) = (1 + e)±(tk" )[CXd + >'4(1 + e)m ±(tk ) task stability is that i) V does not increase between im- ~(1 - e)±(tk )] 50 for ),4 :S 2";(~~e) and Xd = O. Now pacts ii) there is no finite escape time between impacts, during (5 y ) with U = Ue we get: V(tk"+I) - V(tt) = since in this case (5z J reduces to the algebraic equation ILHl (->'2±2(t) - eA3~i(t) - ~FdX(t)t - >'4 F d±(t)t + Fe = Fd. The second condition is readily verified in this >'4FJ)dt->'3Fd [x(t)]::+I+"'¥- [x 2(t)1::+I, hence V(tk+l)one-dof case since all closed-loop equations are linear. 5 - Itt:+l >'2±2(t)dt + >'4Fd It:k +1 x(t)dt + Since after a bounce it may happen that x(t) < -J-L for V(tt) some t, Une and Ut may switch alternatively. We de- >'4Fl[tk+l -tk] (We use the fact that during this period of note Tk the sequence of times when the switches occur, time x < 0). For any tk, tk+l, F d, ±(.) i= 0, one can pick a i.e. X(Tk) = -J-L. Thus on [To,T1] and [T2k +2,T2k+3], small enough >'4 5 >'4 such that the integrand is negative. U = Ut; on [T2k+l, T 2k+2], U = Une , k 2: o. Note This implies that ± decreases. Indeed assume that for a that tk E [T2k. T2k+d. We have: V(t V(To) 5 given k we have V(tk"+I) - V(tt) :S O. This inequality -J-L[>'3Fd + >'1J-L + 2>'IXd)] 50 for 0 < ),3 5 _A,(I':"22: d ). can be rewritten as ~m[±2(tk"+I) -±2(tt)]-CXd[±(tk+1)Now V(tn - V(tk") = ~ V(tk) = (1+ e)±(tk")[CXd - ~(1- ±(tt)] + ~Fl[t~+1 - t~] 5 O. Thus ±2(tk+1) 5 ±2(tt) + e)±(tk")] 50 for c small enough (recall that c can be cho- ~CXd[±(tk"+I) - ±(tt)] - ~Fl[t~+l - t~]. We deduce sen arbitrarily small). However notice that as 1±(tk)1 ap- that for any ),4 > 0, V(tk"+l) - V(tt) 5 0 implies proaches zero, c becomes arbitrarily small for the above I±(tk"+l)I 5 1±(tt)1 when Xd = O. Uniformity of with condition to be fulfilled. A simple solution is to set Xd = 0 respect to k seems difficult to show directly without re(i.e. the desired trajectory is compatible with the con- sorting to explicit integration of the motion as shown straints) so that ~V(tk) 50 independently of c. More- above, since we need to connect the length of [tk, tk+l] over we have been led to add a cross-term in the function and the magnitude of ± in order to conclude. V to get a negative-definite V between impacts. But the • Time-varying Xd(t) calculation of the jump in V, i.e. ~V(tk) reveals that We have seen in the foregoing section that the implethis implies in some sense that Xd must be taken equal mentation of time-varying reference trajectories signifito zero, since otherwise the jump would be positive for cantly modifies the impact phase stability analysis. Ase = 1. Let us now examine V(tk"+I) - V(tt) = [V(tk"+I)sume that Xd is now a twice differentiable time-varying V(T2k+2)] + ~ViT2k+!2 ,T2k + d + [V(T2k+d - V(tt)]· Since signal Xd(t). Then we set U = id - >'2i - ),IX and ne V(tk"+I) - V(T2k+2) = -),3 FdJ-L + ",¥-(x~ - (J-L + Xd)2) Ut = id - ),2i - Fe - ),3Fe. Consider HLC53 with and V(T2k+d - V(tt) = ),3Fd/-L + ",¥-(-x~ + (/-L + Xd)2) >'4 = 0 and >'5 = >'3. For t 2: To, we get between im(these quantities are obtained by integrating V over pacts mi = ->'2i + ),3Fd' We still suppose that the impact phase occurs, i.e. {td is infinite. This is guar2Better results are expected if Ut is replaced by u; = Ut - A2X anteed if i > 0 for t 2: To. In this case we obtain i = _A~Q exp(-~t) + id, ±o < 0, thus a sufficient
f,
P'I
o)-
>..
574
condition is simply Xd > 0 for t ~ To. Proceeding as above we shall analyze V(t k+l ) - V(tt) = Itt:+I (-)..2 + c)i2 (t)dt + [( ~ - ~) x 2(t) + )..3 Fdx (t)] +
suitably chosen, V is such that: • During noncontact phases j = 0, V;: 0 for j ~ l.
::+1
e)..:{d
Itt:+1
x(t)dt, and the variation of V at impact
times, that is given by ~V(tk) = ~m [i(tt)2 - i(tk)2] + C [x(tk)i(tt) - x(tk)i(t k )]· Clearly since e.g. xd(tk+d may be different from Xd(tk) we have to impose further conditions on Xd(t) to guarantee stable impact phase. One of the simplest solution is to guarantee that during the impact phases Xd is constant. Then we know that one of the three controllers above under some conditions on the feedback gains V decreases between impacts i.e. V(tk+l) ::; V(tt) and ~V(tk) = V(tt)- V(t k ) ::; O. Now note first that if Xd > 0, the final value of V when the impact phase has occured stably is V = ~x~. Therefore if we want V to be decreasing " in average", we must choose Xd = 0, otherwise V may take values during the noncontact phase smaller than during the contact phase. Secondly if we want to set Xd = 0 during the impact phase, there is a time To at which we switch the values of xd(To), xd(To), xd(To) to zero and keep those values during the impact and contact phases. We can switch U ne to Ut at the same time. Then we get ~V(To) = V(To+) - V(To-) = mx(TO)xd(To-) + )..IX(TO)Xd(To-) - cxd(To- )xd(To-) + )..IX(TO)Xd(To- ) + CX(TO)Xd(To-) + cxd(To-)x(To) - ~x~(To-) - ~x~(To-)· It is not clear which conditions we should impose to get ~ V(To) ::; o. We shall rather assume that ~ V(To) ~ 0, i.e. V has a positive jump at To. Notice however that if the initial conditions on x ans i are bounded (noncontact phase) then ~ V(To) is bounded as well. Now since after the contact phase we switch U ne again, the closed-loop equation is then mi + )..2i + )..IX = 0, with initial conditions on i and x equal to zero, and this will happen each time the system goes from the contact to the noncontact phase. Thus even if the desired trajectory is "circular" (i.e. the robot is indefinitely in contact and in noncontact phases), we can state that ~V(T6) ::; L < 00 for some L and all j where j E IN denotes the impact phases numbers. For t < To we have already seen that Ut guarantees V(tk+l) ::; V(tt) and ~V(tk) ::; O. A particular feature of the scheme is that V ;: 0 for all t ~ t, (the impacts occur on [to, t,]) except during the transition phase. In summary, from the above studies on stability of diffrerent strategies, we can state the following: consider the one dof example. Assume that H LCS is given by:
= U ne = Ut U = Ut U = Ue U
U
U
=
U ne
at t = 0, x(O) < -J.L t = T6 if x(T6) = -J.L for T6 ::; t ::; Tt for T{ ::; t ::; Td l for Td ::; t ::; Tr
fat
(11)
Consider the function V = ~i2 + ~X2 + cri + ~z~ with Ze = Fe(r)dr . Then if the feedback gains )..1, )..2, )..3, )..4 and the desired trajectories Xd(t) and Fd(t) are
I;
• During contact phases
V ::;
V ::;
-,ne(lxl,lil) for
-,e(lzel), V
= ~z~ .
• During impact phases V(t k+l ) ::; V(tt), ~V(tk) ::; 0, ~V(T6) ::; L < 00. If )..4
= 0 then during contact
phases V ;: 0 and Fe
= o.
Remark. • We get V(t o ) - V(To+) ::; -J.L[)..3Fd + )..1J.Ll, thus V(t o ) ::; -J.L[)..3Fd + )..1J.Ll + ~V(To) + V(To) and clearly there exist )..;, )..j such that )..1 ~ )..j and )..3 ::; )..; imply V(to) ::; V(To-). The values of )..j and )..; may be computed from the expression of ~ V(To) above. • It seems difficult to guarantee that V is always decreasing: indeed firstly the impact phase is not easily stabilizable if Xd is time-varying, and consequently ~V(To) may be positive. Secondly we could use U ne during the impact phase but since Ut or U e have to be switched on, it is preferable to analyze directly what happens with these controllers. Then V cannot be guaranteed to decrease during the periods between impacts. • During the noncontact phases, the closed-loop system's state is (x,i) and it is Ze during contact phases. Another way to formulate this is to say that the system's state is (x, i, ze) with X = i = 0 during contact phases and Ze is constant during noncontact phases. Note that IZe(Td)1 < Iz(t})I, but we may get Iz(t})1 ~ Iz(Tr)l. Since however ~V(T6) is independent of Ze, Ze is guaranteed to remain bounded; we have also to take into account the fact that Ut is switched to U e at T{ E [tk' tk+ll for some k: as we have .seen above for some small enough )..3 and )..4 we get V(t{~I) ::; V(T{) ::; V(t{+). Under these conditions V is decreasing everywhere except during impact phases, i.e. on [T6, t}], but V (t}) ::; V (T6)· Now simple calculations show that when Xd is constant and Ut is applied between two impacts then on [tk. tk+ll, Ix(t)maxl = ;:~t;1, while Ixl does not increase. Thus the amount of increasing of V, say ,k, between two impacts is uniformly bounded in k and -+ 0 as k -+ +00, i.e. ast-+t,. • The switches between tic and tine at Td are not eventbased but rather time-scheduled. Indeed it is not obvious which kind of conditions should be a priori chosen for this switch. In the n-dof case with n ~ 1, one can choose to switch when the manipulator has attained a prespecified 3 domain in the Cartesian space. We now establish global asymptotic stability (not in the Lyapunov sense, but in the sense that all signals are bounded and eventually converge to zero during the contact and noncontact phases) of the overall closed-loop system.
,k
• Lemma Assume V(x)
575
is a positive definite function with
a(llxll) ~ V(x) ~ .6(llxll), a(·) and .60 are class-K functions. Assume that on intervals I j = [TJ, t}], we get along the system's trajectories V(tr) ~ V(Tt-), and V .
umformly bounded on I
j ;
6
let us define n = Ujnj the
complement of I ~ UjIj . Suppose that Il[n] = +00. Then we get the following: i) If on n we have V(x) ~ -')'(llxll) along the system's trajectories and for some class-K function ')'(-), then x(t) -> 0 as t -> +00, tEn. ii) If V(x) ~ -')'I(1lxlll) on n 2j , V(x) ~ -')'2(llx211) on n 2j + 1 , where the system's state is x T =(xf,xn and ll[n 1 ~ Uj n 2j ] = +00, ll[n 2 ~ Uj n 2j ] = +00, then x(t) -> 0 as t -> +00, tEn. iii) If Il[nd < +00, ll[n 2] = +00, and on n 2, Xl(t) is constant and V(x(t)) ~ -')'2(llx211), then limt_+oo X2(t) = o. Proof i) Assume that limt_ooV(x(t)) = 6 > 0 with tEn. Since .6(llxll) 2: V(x) for all t 2: 0 and V is non increasing on n (6 = mintEll V(t)) then we get IIx(t)11 2: .6- 1 (6) > 0 for tEn. Now since tEn we can write for some n (supposing t = 0 E no): V(t) - V(O) = V(t) - V(tj) + V(tj) - V(TO') + V(1Q)V(tj-l) + V(tj-l) - V(T;-l) + V(T;-l) - .... + V(J6l)V(O). Therefore since V(t}) ~ V(Tt) we obtain V(t) y;
V(O) ~ 2:7=1 It~':1 -')'(llx(T)II)dT -
I:
t
It'; ')'(llx(T)ll)dT
-
g')'(lIX(T)ll)dT ~
-,),0.6- 1 (6) 2:7=11l[nj ]. Since for any task Il[n] = +00 we can always find t (and consequently n) such that for any V(O) < +00 and any 6> 0 the inequality V(t) + ')'0.6- 1 (6) 2:7=11l[nj ] ~ V(O) is not verified. Thus we conclude that b = 0, and since Ilx(t)11 ~ a-loV(x(t)) we deduce that limt_+oox(t) = 0 for tEn. If t rf. n then the reasoning fails because for t E In, the term V(t) - V(TO') may be positive. The proof for points ii) and Hi) follow the same lines.
The system we have considered fits within the lemma's framework for suitable choices of the feedback gains, of the desired trajectories and of H LC S. With the notation in (1) we get x T = (zT, z[), Xl = Z, X2 = Ze· Note that the example of stabilization with a unique controller studied in a foregoing section does not. Remark. • It can be proved that the existence of t f < +00 is guaranteed in the example above with ),4 > 0 if V(t k+ l ) ~ V(tt), which in turn guarantees that the conditions of the lemma are fulfilled. 3
4
Conclusions
Vne ' U e and Vc, then form V = aVne + .6Ve , a > 0, .6 > O. Find feedback gains, reference trajectories and H LC S such that solutions to the closed-loop system exist and are unique, such that V is negative definite during non-contact and contact phases, and V(tk+l) ~ V(tt), f:).V(tk) ~ O. Then the closed-loop system fits within the lemma's framework. These ideas can be extended to the case of n-dof manipulators when a linearizing-decoupling control algorithm is applied, since one obtains merely 6 one dof systems in this case.
References [1] B. Brogliato, P. Orhant, "On the transition phase in robotics: impact mod.els dynamics and control" , IEEE Conf. on Robotics and Automation, San Diego, may 1994. [2] B. Brogliato, P. Orhant, R. Lozano, "On the transition phase in robotics- Part 1: impact models and dynamics", IFAC SYROCO'94, Capri, Italy, september 1994. [3] A.F. Filipov, "Differential equations with discontinuous right hand sides" , Kluwer academic publishers, Dordrecht, NL, 1988. [4] Y. Rurmuzlu, T.R. Chang, "Rigid body collisions of a special class of planar kinematic chain", IEEE Trans. on Systems, man, and Cybernetics, vo1.22, no 5, pp. 964-971, september-october 1992. [5] J.K. Mills and D.M. Lokhorst, "Control of robotic manipulators during general task execution: A discontinuous control approach", Int. Journal of Robotics Research, vol.12, no.2, pp.146-163, April 1993. [6] R.P. Paul, "Problems and research issues associated with the hybrid control of force and displacement" , in Proc. IEEE Int. Conf. on Robotics and Automation, pp.1966-1971, 1987. [7] A.1. Panasyuk, "Equations of attainable set dynamics, Part 1: Integral funnel equations", J. of Optimization Theor and Applications, vo1.64, no 2, pp.349-366, 1990. [8] T. Yoshikawa, "Dynamic hybrid position/force control of robot manipl.I.J,jI.tor;:; - Description of hand constraints and calculation of joint driving force", IEEE Trans. on Robotics and Automation, vol.3, no 5, pp.386-392, october 1987.
The general problem of designing a stable control strategy for a complete robotic task can be stated as follows:
f:).k
• Find a transition phase control law Ut such that -> 0, t f < +00, :i:(tk) -> o. Choose U ne and
576
On the transition phase in robotics- Part 2: control B.Brogliato, P. Orhant, R. Lozano* * For the adress of authors please see the companion paper [2]
1
Introduction
This note is devoted to study the stabilization of robot manipulators during complete robotic tasks, i.e. tasks involving both constrained and non-constrained motion phases. It follows [1] [2] where we have stud ied impacts models and dynamics since such tasks invariably imply impacts phenomenons. This kind of problems have received few attention in the robots control literature; among them an interesting work can be found in [5] where the environment is assumed to be compliant. One of our first motivations in doing this work comes from [6], where R.P. Paul stated that The contact problem is unsolved for rigid manipulator, rigid sensor, rigid environment problems. We hope that this work is a first answer. The note is organized as follows : In section 2 we formulate the problem and we study a simple strategy. In section 3 we study more complex controllers that involve switches, and we propose a framework for stabilization of such systems. Conclusions follow.
2
On the control of a "complete" robotic task
• Problem formulation. The problem of the complete robotic task control can be represented by the following set of equations: non-contact : impact:
contact:
. _ (8.) : Z = gne(z, ud, z(O) -
Zo
(t;) the impact; we assume that the environment imposes kinematic constraints on the manipulator that can be written as (z) = O. (8.J represents the motion of the robot's tip along the environment's surface and the dynamic (or algebraic) force error equation, and (8... ,) represents the constrained equation in some suitable coordinates; y(.) may be composed of some components of the task-space velocity X, or of the generalized velocity q. The "high-level control strategy" H LC8 schedules the switches between control laws U ne E U n e and U c E U c (i.e. U = Ui if certain conditions Ci are true. Roughly Zm is the normal component of Z to the constraints. In (1), z(t) may be composed of generalized position and velocity q, q, or of Cartesian coordinates X, X. Then (8.J and (8.,J may be expressed in transformed generalized coordinates, or in task space coordinates [9].Note that the dynamic behaviour of the system considering (8.) and (8 y ) may be quite different from the behaviour considering separately (8.) or (8y ). These difficulties are illustrated in this section. The analysis proceeds in two steps: firstly we study the stability of the bouncing phase, then we examine stability of the constrained phase (if the bouncing phase has occured stably): By stability we mean that the system's state does not diverge and that the robot's tip is eventually at rest on the surface . • computed torque control. Consider the dynamic equations of a rigid manipulator. Let us denote X the vector of position and orientation of the end-effector. Then X = X(q) and X = J(q)q. The generalized forces working on X acting on the end-
. ( (;) effector are impulsive, i.e. A = pOt.. Assuming that (8.) : z =+gne z, Ui, _t). J(q) is square and full-rank, we can apply the control law y y(t k ) = £(y(t k ,phys. para~'b ]Tv, with v = -MJjq + J-TC(q,q)q + J-Tg(q) + { (8 ): . (8.~): Zm = 0 M J (-A2 X -AI(X -Xd )), where M J = J-TMJ-l, Xd is With Y = y(z) the desired value of X . The closed-loop equation is given by: (2)
. . u, = u,(z, x, t, ze) With x = h(z , x , t, control: 1 U, E UncaT"U, E UC { HLC8:u = u, ifC, is true (1) The differential equations (8.) are the dynamical equations for the free motion of the system with and without impacts; the difference equation (8 y ) represents algebraic equations relating the variable value after and before
(tn
Thus we get: zc)
{X;} + A2 X i + Al (Xi - X id )
=0
(3)
where X, denotes the t'th component of X , and: (4)
We now assume that the environment is represented by the kinematic constraints (X) = 0, E lR m , m :S 6.
571
Following Yoshikawa's ideas [9], we split X into two parts as EX = [ EpX. ]
E,X
[
x.P ]. X,
Then the constrained
equation is X, = X, = O. E = [
~;
] is thus a full-
rank matrix that transforms X into task-frame velocity EX. Assume that E = 0 (i.e. the constraint has constant normal vector), i.e. (X) = E,X with E, constant. Then we can write:
A similar analysis can be done when time-varying desired trajectories are considered. However stability may fail in this case since the transition phase including bounces is less easy to stabilize. This indicates the limitations of using a single control law in general. Notice that from EX = EJq and EX(tt) =
Remark. co EX' (t-) k
. Wit
C
h,,= (In0- m -elm 0) co
LI.
duce that (Sy) is Q(tt) (EJ)-I[EJ so that [j with Mj = E- T MjE- 1 = MJ > O. Clearly only X, (the normal component of EX to the surface) has discontinuities due to the impacts, since the surface is supposed to be fri ctionless. Following [4] and since we assume the Jacobian to be full-rank (i.e. the last link is allowed to move in any direction) we deduce that (Sy) is represented . + . . + . - I by X,(tk) = -eX,(tk)' Xp(t k ) = Xp(t k ) . Now the problem is to see whether IX,(t;+I)1 :S IX,(tt)!. Between two impacts we get from (5):
and:
x, + )..2X, +
)..1 (X,
- X'd) = 0
(7)
Thus it can be shown that X, strictly decreases between two impacts (X,(tk+l) = X,(tk) = 0 for all k) and we inwhereas fer that X, and X, tend to zero in finite time XI' and XI' remain bounded during the bouncing phase. (we assume that Xd is such that E,Xd = X,d is "inside " the environment). Thus the impact phase is stable. Let us now examine the closed-loop constrained system: just after and as long as contact is maintained we get:
t"
t,
(8) where Fe is the interaction force expressed in the taskframe coordinates, Fe
= ( F~,
) during constrained
motion, El = rh-m 0], E2 = [0 Im]. Thus Fe! = _(~M;IE'[)-I)..IX!d, and we deduce that all closedloop signals are bounded. Note that the control law in [9] decouples closed-loop dynamics, whereas the computedtorque controller introduces a "disturbance" term in the free-motion equation for XI" It can also be proved that a P D controller does not only guarantee global stability in free motion regulati(m, but also ensures boundedness of the state and the interaction force when the robot collides a rigid environment. 1 Notice
that we are not able to deduce which components of are discontinuous from (4) alone. We need further arguments like the fact that the surface is friction less and Newton's restitution coefficient law . Thus (4) is to be used to compute p from q X rather than the inverse.
EX
(Inom
(-1
)~en Im
ergy is conserved and e <
[jq(t;), with [j (EJ)-I[nEJ, with [n
deLI.
). Clearly whel ' :: = 1 kinetic en-
Iq(tnl
= Iq(t;)I.
1 [n (Inom 0:) then
we can
--+
Notice that if
which indicates that
the bouncing phase (Sy) considered as a discrete time system is convergent. 3
Switching control strategies
3
We now deal with thecase when several control laws are alternatively switched depending on whether the robot is constrained or not. Before going on with a simple example, let us note that when the controller switches from Ui to Uj, the control input may not be defined at the switching time, which in turn may be a switching period. During this time c;>r interval, a reasonnable assumption is that U E [um, UM], where U m and UM may also be time and state dependent. Then the equations in (S%) and (SzJ are differential inclusions. Basic facts and terminology about differential inclusions may be found in [3].
3.1
A simple example
Consider a mass m moving on a horizontal plane with a control force U and a rigid obstacle. Assume that the materials are such that 0 < e < 1. We assume that the controller can a priori switch between three values: • U nc = ->"2 X - >"1 (x - Xd)
A5(Fe - Fd) + >"4 f;o (Fe - Fd)dr - Fe Fd) - Fe with Xd > 0, Fd < 0, and To denotes the switching time from another controller to u c . • Uc
=
•
= >"3(Fe -
Ut
The dynamical equations of the system are given by mx = U in free space, mx = U + it(t) during the impact phase, Fe + U = 0 during the constrained phase. Let us proceed as in [2] to transform the system with distributions into a Caratheodory system. Let us define the "impact" function h(t) where h is of bounded variation on the interval of impacts and it represents the impact impulsive forces (the derivative being taken in distributional sense) and f(t) such that f is a .e. differentiable but at the points of discontinuities of h, with j = h outside these points. Since changes of variables
572
To < tf (Le. the bounces have not stopped), and that To coincides with an impact. The closed-loop equation is:
are allowed in differential inclusions [3] (p.99, th.l), we set Y
=(
~~
) =(
~
{m ).
) - (
Then we get:
mi (
~~
) = (
~
)
+(
~
{~~~
A4 J~ Fddt
= -A5Fd -
A4Fdt + h(t)
+ h(t) (10)
)
(9)
Thus we have transformed the original system with distributions in coefficients into a differential inclusion of the form iJ E F(y, u) = Ay + Bu, with U E [um(y, t), UM(y, t)] at the switching times, U given by one of the controllers above between two switches. We can consider um(y, To) = min(u(To-), u(To+)), UM(y, To) = max(u(To-), u(To+)). The local existence and uniqueness theorems in [7] for the nonautonomous case require that the multifunction be measurable in t . We deduce that provided both functions um(Y, t) and UM(y, t) are measurable in t, the general theorems above can be applied to (9) for any initial data outside the discontinuities of h(t) (i.e. they can be applied almost everywhere with h as above). Note that if the solutions y(t) to the inclusion in (9) exist and are unique (this is to be understood for reachable sets) on a certain interval [to, t], so do the original trajectories X(t) = y(t) - (
= -A5Fd -
).
• TI-ansition phase stability with H LC5} Let us consider our example. Assume we choose the following strategy H LC5}: After the first impact has occured switch to Ut (If the mass is still bouncing, then Ut = -A3Fd). Then after a certain prespecified time switch to U c . Notice that we may assume in a first study that the switches occur infinitely fast. Now we can reasonnably assume that the bounds um(Y, t) and UM(Y, t) are measurable in t, because each time a switch occurs, both controllers before and after the switch lead to an equation in (9) with a time-measurable RHS, since Unc = unc(y}.!(t)) when the switch between Unc and Ut occurs. At a given time t, either one of the above controllers is used, in which case F(t, y) is a point in the plane and (9) is an ordinary differential equation for which Carath60dory conditions trivially holds, or t corresponds to a switching instant, and F(t, y) is a segment, thus convex. The only case of interest is when a transition between U nc and another controller occurs, since otherwise Urn and UM can be assumed to be independent of y. In this case one can verify that o:(F(t,y),F(t,y')) has the form «Y2 - y;)2 + (UM - u~ )2)i, where UM can also be replaced by U m depending on the values of the bounds. Thus o:(F(t,y),F(t,y')) ~II y - y' 11 provided the bounds are smdoth enough functions of y, so that theorem 2.2 in [7] applies. Then if we prove that no finite escape can occur in the system, the solutions (Le. the reachable sets) from any admisssible initial data exist and are unique on [0, +00). The controllers Unc and Ut both lead a stable transition phase. Assume now that for some reason U c is switched at t = To (we take To = 0 to simplify the analysis), with
Between two jumps we get mi = "-A5Fd - A4Fdt, or i = ·at + b with a = _A.Fd > 0, b = -~ > o. Thus m m ± = + bt + c, with c = ±(O+) < o. Note that ± is increasing and ±(t) > 0 for some t, so that the mass collides the wall again at the time t} = ~ ( -~ + ~),
af
t!. = ~ - ~a±(O+) > 0, and ±(t;-) = ~ti + bt} + ±(O+) > ±(O+). Note that t} -+ as a -+ o. We want that 1±(ti)1 = e±(t;-) < I±(O+)I· This condition can be written as ~ti+bt} < I±(O+)I~. When A4 -+ 0, it becomes simply e < 1. We deduce that there exists A4 > 0 such that for 0 < A4 < A4 the condition is fulfilled. We have asssumed that To coincides with a percussion, so that x(O) = 0 and we can explicitely calculate t}. Note however that if such is not the case, we can still assert existence of tJ (because there exists tJ such that for t > tJ, ±(t) > 0), and of A; as it can be shown that A4 = 0 implies a stable transition phase (for any A5 > 0, there exists A4 > 0 such that A4 E (0, A4) yields 1±(ti)1 < I±(O+)I) . Now we can repeat this reasoning for intervals [tJ, t2], etc... by setting each time tk = 0 and integrating between tk and tk+l. It can also be proved that the values of A4 on the successive intervals are in some sense independent of k. We conclude that there exists A4 > 0 such that the impact phase with U = U c is stable. There is a clear reason why adding an integral force feedback introduces some difficulties in the analysis of the transition phase behaviour. In effect, by combining (5z ) and (511 ) when Ut is used we are able to get the following recurrence equations for ±(tk): ±(tt+J) =
-¥
-e [29a
(-~ + ~r + ~ (-~ +~) + ±(tn] ,
for
which 0 si an equilibrium point. The only thing that we have been able to conclude is that for A4 small enough, these recurrence equation lead for any initial condition to t!.k -+ 0 and :i:(tk) -+ 0 as k -+ +00. • TI-ansition phase stability with H LC52 We assume now that the switch between U nc and Ut is triggered when x(t) = -j..I. ~ 0, and U c is switched on when F. > o. We assume that the force sensor response to impacts is such that after each percussion, U c is switched on during a certain period of time t!.c. Then it can be shown that there exists A4 > 0 such that for o < A4 < A4, H LC52 leads to a stable task whatever t!.c may be.
3.2
A Lyapunov-like stability analysis
In view of the extension of the preceding results to the case of n-dof manipulators, it is interesting to estab-
573
lish stability using Lyapunov functions V. First notice that since we have guaranteed existence of trajectories (see [2]), Lyapunov arguments can be used to prove stability. Roughly, when a jump occurs in the solution, there corresponds a jump .6.v(tk) in V . We thus have to analyze the continuous variation and the discontinuous variation of V. We shall require the use of a single Lyapunov V function to prove stability of a whole task as described in (1) . The main difficulty will be to guarantee negative V and ~ V (tk) along all trajectories of (1) , knowing that at certain times, a "wrong" controller may be applied. Let us for the moment investigate the one-dof case, and consider the function
(T2k+2, tk"+I) and (tt, T 2k+d respectively), and since ~ViT"+2,T,.+d 5 0 we deduce that V(tt+l) 5 V(tt). In conclusions, V is negative definite during the free motion 5 V(To) and V(tt+l) 5 V(tt) during the phase, V(t impact phase so that V (t f) :S V (To) (t f is the time when bounces stop, see example 6 above), and Fe = Fd < 0 for
o)
t 2: tf· Another strategy H LC 53 may be to apply Ut for all t 2: To. In this case, we only have to verify that between impacts V decreases. We get: V(tk"+I) - V(tt) = -),3Fd [x(t)]::+1 +),1 [x(t)21:: +1 +(C-),2) I/:+I ±2(t)dt-
e>-;';d I/:+l x(t)dt 5 0 since XFd > 0 on [tt, tk"+I] and >'2 > C. V = ~m±2 + (x - Xd)2 + c:i:x + ~ (J~ Fe(T)dT • integml force feedback with c > 0 satisfying c2 < m),I ' Suppose that ),4 = 0, We now briefly analyze the case when ),4 > 0, i.e. there is ),5 = ),3, Ut = ),3(Fe - Fd) - Fe - ),2± (we add dampsome integral action in the force feedback, and the strating in the transition control law 2) and that H LC5 = egy consists of switching between Une and Ue at To. We do H LC52 . Along trajectories of (5z ) with U = U ne we get the hypothese that Fd == 0 whenever U i= U e . Thus along V 5 (->'2 +c+ ~)±2 + (-~c +~) x2 , thus V is neg- trajectories of (5z ) with U = Une, V is negative definite ative definite for ),2 > C + ~ , ),1 > ~ . We implicitely in X, ± since I~ FddT is constant during that time. The assume that the impact phase effectively occurs, i.e. {td first variation of V to analyze is that at the impact times; 2 is an infinite sequence. Then a sufficient condition for the we get: ~ V(tk) = (1 + e)±(tk" )[CXd + >'4(1 + e)m ±(tk ) task stability is that i) V does not increase between im- ~(1 - e)±(tk )] 50 for ),4 :S 2";(~~e) and Xd = O. Now pacts ii) there is no finite escape time between impacts, during (5 y ) with U = Ue we get: V(tk"+I) - V(tt) = since in this case (5z J reduces to the algebraic equation ILHl (->'2±2(t) - eA3~i(t) - ~FdX(t)t - >'4 F d±(t)t + Fe = Fd. The second condition is readily verified in this >'4FJ)dt->'3Fd [x(t)]::+I+"'¥- [x 2(t)1::+I, hence V(tk+l)one-dof case since all closed-loop equations are linear. 5 - Itt:+l >'2±2(t)dt + >'4Fd It:k +1 x(t)dt + Since after a bounce it may happen that x(t) < -J-L for V(tt) some t, Une and Ut may switch alternatively. We de- >'4Fl[tk+l -tk] (We use the fact that during this period of note Tk the sequence of times when the switches occur, time x < 0). For any tk, tk+l, F d, ±(.) i= 0, one can pick a i.e. X(Tk) = -J-L. Thus on [To,T1] and [T2k +2,T2k+3], small enough >'4 5 >'4 such that the integrand is negative. U = Ut; on [T2k+l, T 2k+2], U = Une , k 2: o. Note This implies that ± decreases. Indeed assume that for a that tk E [T2k. T2k+d. We have: V(t V(To) 5 given k we have V(tk"+I) - V(tt) :S O. This inequality -J-L[>'3Fd + >'1J-L + 2>'IXd)] 50 for 0 < ),3 5 _A,(I':"22: d ). can be rewritten as ~m[±2(tk"+I) -±2(tt)]-CXd[±(tk+1)Now V(tn - V(tk") = ~ V(tk) = (1+ e)±(tk")[CXd - ~(1- ±(tt)] + ~Fl[t~+1 - t~] 5 O. Thus ±2(tk+1) 5 ±2(tt) + e)±(tk")] 50 for c small enough (recall that c can be cho- ~CXd[±(tk"+I) - ±(tt)] - ~Fl[t~+l - t~]. We deduce sen arbitrarily small). However notice that as 1±(tk)1 ap- that for any ),4 > 0, V(tk"+l) - V(tt) 5 0 implies proaches zero, c becomes arbitrarily small for the above I±(tk"+l)I 5 1±(tt)1 when Xd = O. Uniformity of with condition to be fulfilled. A simple solution is to set Xd = 0 respect to k seems difficult to show directly without re(i.e. the desired trajectory is compatible with the con- sorting to explicit integration of the motion as shown straints) so that ~V(tk) 50 independently of c. More- above, since we need to connect the length of [tk, tk+l] over we have been led to add a cross-term in the function and the magnitude of ± in order to conclude. V to get a negative-definite V between impacts. But the • Time-varying Xd(t) calculation of the jump in V, i.e. ~V(tk) reveals that We have seen in the foregoing section that the implethis implies in some sense that Xd must be taken equal mentation of time-varying reference trajectories signifito zero, since otherwise the jump would be positive for cantly modifies the impact phase stability analysis. Ase = 1. Let us now examine V(tk"+I) - V(tt) = [V(tk"+I)sume that Xd is now a twice differentiable time-varying V(T2k+2)] + ~ViT2k+!2 ,T2k + d + [V(T2k+d - V(tt)]· Since signal Xd(t). Then we set U = id - >'2i - ),IX and ne V(tk"+I) - V(T2k+2) = -),3 FdJ-L + ",¥-(x~ - (J-L + Xd)2) Ut = id - ),2i - Fe - ),3Fe. Consider HLC53 with and V(T2k+d - V(tt) = ),3Fd/-L + ",¥-(-x~ + (/-L + Xd)2) >'4 = 0 and >'5 = >'3. For t 2: To, we get between im(these quantities are obtained by integrating V over pacts mi = ->'2i + ),3Fd' We still suppose that the impact phase occurs, i.e. {td is infinite. This is guar2Better results are expected if Ut is replaced by u; = Ut - A2X anteed if i > 0 for t 2: To. In this case we obtain i = _A~Q exp(-~t) + id, ±o < 0, thus a sufficient
f,
P'I
o)-
>..
574
condition is simply Xd > 0 for t ~ To. Proceeding as above we shall analyze V(t k+l ) - V(tt) = Itt:+I (-)..2 + c)i2 (t)dt + [( ~ - ~) x 2(t) + )..3 Fdx (t)] +
suitably chosen, V is such that: • During noncontact phases j = 0, V;: 0 for j ~ l.
::+1
e)..:{d
Itt:+1
x(t)dt, and the variation of V at impact
times, that is given by ~V(tk) = ~m [i(tt)2 - i(tk)2] + C [x(tk)i(tt) - x(tk)i(t k )]· Clearly since e.g. xd(tk+d may be different from Xd(tk) we have to impose further conditions on Xd(t) to guarantee stable impact phase. One of the simplest solution is to guarantee that during the impact phases Xd is constant. Then we know that one of the three controllers above under some conditions on the feedback gains V decreases between impacts i.e. V(tk+l) ::; V(tt) and ~V(tk) = V(tt)- V(t k ) ::; O. Now note first that if Xd > 0, the final value of V when the impact phase has occured stably is V = ~x~. Therefore if we want V to be decreasing " in average", we must choose Xd = 0, otherwise V may take values during the noncontact phase smaller than during the contact phase. Secondly if we want to set Xd = 0 during the impact phase, there is a time To at which we switch the values of xd(To), xd(To), xd(To) to zero and keep those values during the impact and contact phases. We can switch U ne to Ut at the same time. Then we get ~V(To) = V(To+) - V(To-) = mx(TO)xd(To-) + )..IX(TO)Xd(To-) - cxd(To- )xd(To-) + )..IX(TO)Xd(To- ) + CX(TO)Xd(To-) + cxd(To-)x(To) - ~x~(To-) - ~x~(To-)· It is not clear which conditions we should impose to get ~ V(To) ::; o. We shall rather assume that ~ V(To) ~ 0, i.e. V has a positive jump at To. Notice however that if the initial conditions on x ans i are bounded (noncontact phase) then ~ V(To) is bounded as well. Now since after the contact phase we switch U ne again, the closed-loop equation is then mi + )..2i + )..IX = 0, with initial conditions on i and x equal to zero, and this will happen each time the system goes from the contact to the noncontact phase. Thus even if the desired trajectory is "circular" (i.e. the robot is indefinitely in contact and in noncontact phases), we can state that ~V(T6) ::; L < 00 for some L and all j where j E IN denotes the impact phases numbers. For t < To we have already seen that Ut guarantees V(tk+l) ::; V(tt) and ~V(tk) ::; O. A particular feature of the scheme is that V ;: 0 for all t ~ t, (the impacts occur on [to, t,]) except during the transition phase. In summary, from the above studies on stability of diffrerent strategies, we can state the following: consider the one dof example. Assume that H LCS is given by:
= U ne = Ut U = Ut U = Ue U
U
U
=
U ne
at t = 0, x(O) < -J.L t = T6 if x(T6) = -J.L for T6 ::; t ::; Tt for T{ ::; t ::; Td l for Td ::; t ::; Tr
fat
(11)
Consider the function V = ~i2 + ~X2 + cri + ~z~ with Ze = Fe(r)dr . Then if the feedback gains )..1, )..2, )..3, )..4 and the desired trajectories Xd(t) and Fd(t) are
I;
• During contact phases
V ::;
V ::;
-,ne(lxl,lil) for
-,e(lzel), V
= ~z~ .
• During impact phases V(t k+l ) ::; V(tt), ~V(tk) ::; 0, ~V(T6) ::; L < 00. If )..4
= 0 then during contact
phases V ;: 0 and Fe
= o.
Remark. • We get V(t o ) - V(To+) ::; -J.L[)..3Fd + )..1J.Ll, thus V(t o ) ::; -J.L[)..3Fd + )..1J.Ll + ~V(To) + V(To) and clearly there exist )..;, )..j such that )..1 ~ )..j and )..3 ::; )..; imply V(to) ::; V(To-). The values of )..j and )..; may be computed from the expression of ~ V(To) above. • It seems difficult to guarantee that V is always decreasing: indeed firstly the impact phase is not easily stabilizable if Xd is time-varying, and consequently ~V(To) may be positive. Secondly we could use U ne during the impact phase but since Ut or U e have to be switched on, it is preferable to analyze directly what happens with these controllers. Then V cannot be guaranteed to decrease during the periods between impacts. • During the noncontact phases, the closed-loop system's state is (x,i) and it is Ze during contact phases. Another way to formulate this is to say that the system's state is (x, i, ze) with X = i = 0 during contact phases and Ze is constant during noncontact phases. Note that IZe(Td)1 < Iz(t})I, but we may get Iz(t})1 ~ Iz(Tr)l. Since however ~V(T6) is independent of Ze, Ze is guaranteed to remain bounded; we have also to take into account the fact that Ut is switched to U e at T{ E [tk' tk+ll for some k: as we have .seen above for some small enough )..3 and )..4 we get V(t{~I) ::; V(T{) ::; V(t{+). Under these conditions V is decreasing everywhere except during impact phases, i.e. on [T6, t}], but V (t}) ::; V (T6)· Now simple calculations show that when Xd is constant and Ut is applied between two impacts then on [tk. tk+ll, Ix(t)maxl = ;:~t;1, while Ixl does not increase. Thus the amount of increasing of V, say ,k, between two impacts is uniformly bounded in k and -+ 0 as k -+ +00, i.e. ast-+t,. • The switches between tic and tine at Td are not eventbased but rather time-scheduled. Indeed it is not obvious which kind of conditions should be a priori chosen for this switch. In the n-dof case with n ~ 1, one can choose to switch when the manipulator has attained a prespecified 3 domain in the Cartesian space. We now establish global asymptotic stability (not in the Lyapunov sense, but in the sense that all signals are bounded and eventually converge to zero during the contact and noncontact phases) of the overall closed-loop system.
,k
• Lemma Assume V(x)
575
is a positive definite function with
a(llxll) ~ V(x) ~ .6(llxll), a(·) and .60 are class-K functions. Assume that on intervals I j = [TJ, t}], we get along the system's trajectories V(tr) ~ V(Tt-), and V .
umformly bounded on I
j ;
6
let us define n = Ujnj the
complement of I ~ UjIj . Suppose that Il[n] = +00. Then we get the following: i) If on n we have V(x) ~ -')'(llxll) along the system's trajectories and for some class-K function ')'(-), then x(t) -> 0 as t -> +00, tEn. ii) If V(x) ~ -')'I(1lxlll) on n 2j , V(x) ~ -')'2(llx211) on n 2j + 1 , where the system's state is x T =(xf,xn and ll[n 1 ~ Uj n 2j ] = +00, ll[n 2 ~ Uj n 2j ] = +00, then x(t) -> 0 as t -> +00, tEn. iii) If Il[nd < +00, ll[n 2] = +00, and on n 2, Xl(t) is constant and V(x(t)) ~ -')'2(llx211), then limt_+oo X2(t) = o. Proof i) Assume that limt_ooV(x(t)) = 6 > 0 with tEn. Since .6(llxll) 2: V(x) for all t 2: 0 and V is non increasing on n (6 = mintEll V(t)) then we get IIx(t)11 2: .6- 1 (6) > 0 for tEn. Now since tEn we can write for some n (supposing t = 0 E no): V(t) - V(O) = V(t) - V(tj) + V(tj) - V(TO') + V(1Q)V(tj-l) + V(tj-l) - V(T;-l) + V(T;-l) - .... + V(J6l)V(O). Therefore since V(t}) ~ V(Tt) we obtain V(t) y;
V(O) ~ 2:7=1 It~':1 -')'(llx(T)II)dT -
I:
t
It'; ')'(llx(T)ll)dT
-
g')'(lIX(T)ll)dT ~
-,),0.6- 1 (6) 2:7=11l[nj ]. Since for any task Il[n] = +00 we can always find t (and consequently n) such that for any V(O) < +00 and any 6> 0 the inequality V(t) + ')'0.6- 1 (6) 2:7=11l[nj ] ~ V(O) is not verified. Thus we conclude that b = 0, and since Ilx(t)11 ~ a-loV(x(t)) we deduce that limt_+oox(t) = 0 for tEn. If t rf. n then the reasoning fails because for t E In, the term V(t) - V(TO') may be positive. The proof for points ii) and Hi) follow the same lines.
The system we have considered fits within the lemma's framework for suitable choices of the feedback gains, of the desired trajectories and of H LC S. With the notation in (1) we get x T = (zT, z[), Xl = Z, X2 = Ze· Note that the example of stabilization with a unique controller studied in a foregoing section does not. Remark. • It can be proved that the existence of t f < +00 is guaranteed in the example above with ),4 > 0 if V(t k+ l ) ~ V(tt), which in turn guarantees that the conditions of the lemma are fulfilled. 3
4
Conclusions
Vne ' U e and Vc, then form V = aVne + .6Ve , a > 0, .6 > O. Find feedback gains, reference trajectories and H LC S such that solutions to the closed-loop system exist and are unique, such that V is negative definite during non-contact and contact phases, and V(tk+l) ~ V(tt), f:).V(tk) ~ O. Then the closed-loop system fits within the lemma's framework. These ideas can be extended to the case of n-dof manipulators when a linearizing-decoupling control algorithm is applied, since one obtains merely 6 one dof systems in this case.
References [1] B. Brogliato, P. Orhant, "On the transition phase in robotics: impact mod.els dynamics and control" , IEEE Conf. on Robotics and Automation, San Diego, may 1994. [2] B. Brogliato, P. Orhant, R. Lozano, "On the transition phase in robotics- Part 1: impact models and dynamics", IFAC SYROCO'94, Capri, Italy, september 1994. [3] A.F. Filipov, "Differential equations with discontinuous right hand sides" , Kluwer academic publishers, Dordrecht, NL, 1988. [4] Y. Rurmuzlu, T.R. Chang, "Rigid body collisions of a special class of planar kinematic chain", IEEE Trans. on Systems, man, and Cybernetics, vo1.22, no 5, pp. 964-971, september-october 1992. [5] J.K. Mills and D.M. Lokhorst, "Control of robotic manipulators during general task execution: A discontinuous control approach", Int. Journal of Robotics Research, vol.12, no.2, pp.146-163, April 1993. [6] R.P. Paul, "Problems and research issues associated with the hybrid control of force and displacement" , in Proc. IEEE Int. Conf. on Robotics and Automation, pp.1966-1971, 1987. [7] A.1. Panasyuk, "Equations of attainable set dynamics, Part 1: Integral funnel equations", J. of Optimization Theor and Applications, vo1.64, no 2, pp.349-366, 1990. [8] T. Yoshikawa, "Dynamic hybrid position/force control of robot manipl.I.J,jI.tor;:; - Description of hand constraints and calculation of joint driving force", IEEE Trans. on Robotics and Automation, vol.3, no 5, pp.386-392, october 1987.
The general problem of designing a stable control strategy for a complete robotic task can be stated as follows:
f:).k
• Find a transition phase control law Ut such that -> 0, t f < +00, :i:(tk) -> o. Choose U ne and
576